An Iraqi Shi'ite force backed by Iran has said it pushed Islamic State (IS) militants out of villages on the border with Syria on May 29, in a step toward reopening a supply route to send Iranian weapons to President Bashar al-Assad.

The Popular Mobilization Forces described its advance to the border with Syria north of the town of Baaj on May 29 as "a Ramadan miracle," referring to the Muslim fasting month that started over the weekend.

For the Popular Mobilization Forces, reclaiming the territory is a step toward achieving a linkup between Baghdad and Assad's forces in Syria, reopening a critical highway for supplying weapons to Assad's army, and giving the Syrian leader a significant advantage in fighting the 6-year-old armed rebellion against him.

"This will be the first step to the liberation of the entire border," said Ahmad al-Asadi, a spokesman for the Iranian-backed militia.

But the Iraqi forces backed by Iran cannot as yet link up with the Syrian Army and allied Iranian-backed militias in Syria, as Assad's forces have not as yet reached the Iraqi border from the Syrian side, despite concerted efforts in recent weeks.

To get there, they would have to pass through territory held by U.S.-backed Syrian Kurdish forces that the U.S. military has been training to fight IS.

Iraq's Iranian-backed militia leaders say they are ready to move inside Syrian territory to assist Assad in battling rebel groups there. It is not known whether the Syrian Kurds would allow the Iraqi Shi'ite force to use their territory to reach Assad's troops, however, which are deployed further south and further west.

The Popular Mobilization Forces are taking part in the U.S.-backed Iraqi campaign to defeat IS in its Iraqi stronghold of Mosul and surrounding Nineveh Province.

The Iraq government's army has been focusing its efforts on dislodging the militants from Mosul, while the Iranian-backed forces have battled to reclaim the vast desert territory between Mosul and the Syrian border.

While reporting nominally to Iraq's Shi'ite-led government, the Popular Mobilization Forces have Iranian military advisers, one of whom died last week fighting near Baaj.

Iran backs militia forces in both Iraq and Syria, where it has helped to train and organize thousands of Shi'ite troops it has recruited from Iraq, Afghanistan, and Pakistan. Fighters from Lebanon's Hizballah also work closely with Iranian military commanders in Syria.

Eight months into the Mosul campaign, IS fighters have been dislodged from all of the city except an enclave by the western bank of the Tigris River.

Iraq's army over the weekend launched a new offensive to take the militants' enclave, which includes the densely populated Old City.